How can a mechanical linkage with a closed loop be rigged?
There are generally two ways here: bone constraints and rigidbody simulation.

Solution 1 - Bone Constraints:




Create the linkages (meshes).
Create armature with bones set as shown on the image below.
Animate the key bone (the red one here).
Rigging.






Result:



Example file: 

Solution 2 - Rigidbody Simulation:




Build the initial state of the linkage, make sure they don't intersect or too close to each other;
Select them all, click Add Active button in Physics tab in Toolshelf;
Select either two linkages, click Connect in that tab to create constraints, set type to Hinge, move them to expected hinge points. Note that Hinge rotates along the Z axis of constraints (empty objects), so you may need to correct the rotation.
Create rotation action for BoneEB (the blue bone in the image below), toggle Animated in its Rigid Body setting panel.







For better result, you need to set the Steps per Second and Solver Iterations relatively high in Scene tab.

Result:



Example file: